This post discusses the concept of rate of return using adjusted close stock prices with both monthly and annual compounding.
Question: The table below has the adjusted closing
price of Vanguard’s emerging market fund (VEIEX) at two dates  the date of the fund’s inception and the
opening date of 2016. What is the
annualized return on this fund assuming that returns are compounded
monthly?
Data:
Price of Vanguard
Emerging Market Price
at Two Dates


Date

Adjusted Close Price
of VEiEX

5/4/94

7.0

1/4/16

19.3

The data used here was also used in a post on whether the
VEIEX fund is a suitable product for investors attempting to diversify and
minimize risk associated with the U.S. market. That post can be found here.
Analysis:
The monthly rate of return r that leads to VEIEX going from
7.0 to 19.3 months over this time span can be obtained from the equation
F=S x (1+r_{m})^{n }
Where F is 19.3, S is 7.0 and n is the number of elapsed
months.
I need to find the number of elapsed months.
To do this I went back to the original database with all
data on all months, created a column variable called month, and numbered the months
from 0 to n.
The first and last row of this database looks like this
VEIEX Price Data with
Month Column


Date

Month

Adjusted Close Price of
VEiEX

5/4/94

0

7.0

1/4/16

260

19.3

Here 260 is the number of elapsed months.
Note on Calculation
of Elapsed Months: The number of
elapsed months could also be calculated by placing date information into the
month and year functions. Interested
reader should go here.
So solve for the rate
of return
r_{m} = (19.3/7.0) ^{(1/260)}  1
=0.003908
annualize by multiplying by 12 and get
0.0469
So our answer is 4.69%.
A Technical Note:
The annual interest rate calculated here of 4.69% is the
interest rate when interest is compounded monthly. What if interest was not compounded monthly? What if it was compounded annually?
260 months is the same as 21.67 years.
The annual interest rate compounded annually is
r = (19.3/7)^{(1/21.67}) 1
or
4.792 %
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